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I. Mariotte^s and Gay-Ltissac's laics. — Gases and superheated vapors 
tend towards a limiting state, called that of a perfect gas, which is char- 
acterized by the two following laws : 

1. Mariotte's laic. — The pressures of the same mass of gas are inversely 
proportional to the volumes. 

2. Gay-Lussac's law. — All gases have the same co-efficient of dilata- 
tion under constant pressure, and this co-efficient is independent of the 

These two laws are expressed by the equation, 

(1) 'pV = l^(l-\. at)) 

where v is the volume of unit of weight of the gas, p the pressure^ t 
the temperature, a the co-efficient of dilatation nnder constant pressure, 
and K a constant depending upon the nature of the gas. 

2. Absolute temperature. — The coefficient of dilatation a, which is the 
same for all perfect gases according to Gay-Lussac's law, is grs"? nearly, 
according to the experiments of Regnault. Consequently, equation (1) 

(2) ..".,... ^. = A(273+.). 

The factor (273 + t) is called the absolute temperature in the dynamic 
tbeory of heat. It is the centigrade temperature counted from a zero 
placed 273 degrees below the ordinary zero. 

3. Specific volume. — If we call ^o the normal atmospheric pressure, and 
put K = ^0 ^'o^ equation (2) may be written, 

(3) . ^ jp« =^1^(273 + «), 

and, in this form, we see that Vq is the value of v which corresponds to 
the values f = o and p = p(^oi the temperature^ and pressure. 

This constant, which is called the specific volume, represents then 
the volume of unit of weight of the gas at zero temperature, and, under 
the normal pressure ; if the gas underconsideration can reach these con- 
ditions of pressure and temperature without change of state and with- 
out ceasing to satisfy Mariotte's and Gay-Lussac^s laws ; its numerical 
value is the reciprocal of the specific weight, or weight of the unit of 
volume at zero and under the pressure po. 

The specific volume ceases evidently to have this signification for gases 
and vapors which are in the state of a perfect gas only at temperatures 
above zero. Its definition results from equation (3), which the gas un- 
der consideration must satisfy in the limiting state ; its numerical value 
may be derived from that equation by determining by experiment, in 
the physical conditions where the equation is applicable, a system of 
values corresponding to p, v, and t. 

The determination of specific volumes, or, what is the same thing, the 
densities of gases and vapors, is of, great importance in chemistry : aad 
they have been the object of the research of many eminent scientists. 
We shall return, at the end of this chapter, to the physical laws which 
have been establi^ed, dwelling particularly upon those which are use- 
ful in the approximate calculation of the force of explosive substances. 

4. Specific heats. — The specific heat of a substance is the quantity of 
heat necessary to raise unit of weight of the substance one degree 
centigrade. The quantity may be measured in two ways; the body which 
is being heated may be allowed to expand freely under a determined 
pressure, or the volumeof the body may be maintained constant; in the 
first case the specific heat is of constant pressure, and in the second of 
constant volume. 

Calorimetric experiments have established the following law, which, 
like those of Mariotte and G-ay-Lussac, appears to characterize the 
state of the perfect gas: 

The specific heats under constant pressure and constant volume are inde- 
pendent of the pressure and volume. 

This law has been experimentally verified by Eegnault by^the direct 
determination of specific heats under constant pressure. 

As to the specific heats for constant volume, their direct measure- 
ment being almost impossible, this verification cannot be made. -The 
hxw appears, however, sufficiently confirmed by the indirect measure- 
ment which has been made, of the relation between the specific heats 
throughout a considerable range of temperature and pressure, by many 

5. Conversion of heat into work by the expansion of gas. — The abos^e 
laws being established, we may deduce as follows the relation that ex- 
ists between the variations of volume and pressure of a gas and the 
ieat necessary to produce them. 

Resuming the fandameiital equation (3), and writing for .shortness, 

(*) « = 2^1" 

and, calling T the absolute temperature, 273 + /, this equation may be 


(5) p V = ¥.T. 

The consequences which follow are these : 

1st. If. the pressure, p. remaining constant, the volume varies by the 
amount d v, the temperature undergoes a corresponding variation rep- 
resented by ^-p— ; and, consequently, the gas has received a quantity 

of heat ^^ — , & being the specific heat under constant pressure. 
2d. If. the volume, r, remaining constant, the , >ressure varies by d p, 

the temperature varies by r^^ : and the gas has received a quantity of 

c V d p 
heat equal to p-^. c being the specific heat under constant volume. 

3d. Consequently, if the volume and pressure increase together by 
dv and dp. the gas receives a quantity of heat ; 

(6) dq=^[c' p dv -f c vdp). 

Differentiating (5), we have, 

(7) ^dT=pdr + vdp, 

and. eliminating successively between (6) and (7) dp and dv. we have 
the two equations, 

(8) dq = cdT -^^ ~^pdv. 

(9) dq = &dT -^~^-^vdp, 

which, together with (6). contain all the thermodynamic laws of 

6. ^Ye will consider first equation (8). The quantity j> f7r, which ap- 
pears in its second member, is evidently the work done by the elastic 
force of the gas when the volume increases by dv. 

Let w be an infinitely small element of the surface which incloses 
the gas. The pressure on this element is }) w. since p is the pressure on 
unit of surface : and the element of work of this pressure correspond- 
ing to an infinitely small increase of volume is p w h. h being the dis- 
placement of w perpendicular to itself. The work done then is j^ I w h ; 
but I oj k is the variation of the volume dv ; therefore j; dv is the external 
work done by the gas. 

It results, then, from equation (8) that the quantity of heat absorbed 

by a gas in an infinitely small change is composed of two terms j of 
which the first is proportional to the change of temperature, and the 
second to the element of external work. 

If we consider a change which alters by finite quantities the tempera- 
ture and volume of a gas, we derive, by integrating (8), the quantity of 
lieat absorbed during the change, 

^ fCc d T +^-^ P dv}) 

If c and e' are functions of T and v, this expression can only be inte- 
grated if we know the relation between them ; but, if we assume (No. 4) 
that c and c' are constants, we have, 


or calling To and Ti the initial and final temperatures and 6 the total 
external work done by the elastic force of the gas, we have 

(10) ^ = c(Tf-To) +^^^. 

6J. Equivalence of the heat and worJc. — If we put Ti = To in the last 
equation ; that is, if the gas returns to its initial state, we have, 

(11) 1=-^'- 

Thus, in this case, the amount of heat absorbed by the gas is pro- 
portional to the external work done. 
The quantity 

(12) A = 


which expresses the ratio between the heat absorbed and the work done, 
is called the calorific equivalent of the work ; and its reciprocal, 

03) E = ^-^^ 

is the mechanical equivalent of the heat. . 

We thus see that we may deduce from the known laws which govern 
^ses the notion of the equivalence of heat and work, whose precise 
conception has led to so great progress in the theory of heat. The funda- 
mental postulate of this theory consists in the assertion that the quan- 
tity which has been designated E is invariable, and independent of the 
nature of the body which has served as intermediate in the transforma- 
tion of heat into work. We need not recall here the reasoning which 

has established the soundness of this principle, which may 'otherwise 
be considered as sufficiently confirmed by the verification of its numer- 
ous consequences. 

We consider, then, that, for perfect gases, there exists between the 
volume and the two specific heats the relation expressed by (13) ; or, 
taking into consideration the value of R (4), the relation, 

<^^) ^ = 273 &:^c 

expresses an invariable number, depending solely upon the choice of 
the units which serve to express the quantities of work and heat. 

7. Value of the mechanical equivalent of heat. — Since the value of E is 
independent of the nature of the gas, we may derive it from the values 
of ^0, c', and c, which have been experimentally determined for many 
gases, following the laws of Mariotte and Gay-Lussac. We take, for 
example, atmospheric air. 

Regnault's experiments show that under constant pressure the spe- 
cific heat of air is 

& = .23754, 

and that its specific volume (the reciprocal of the specific weight), 
taking for units the meter and the kilogram, is 

_ ^ 

Finally, the ratio between the two specific heats, deduced from the ve- 
locity of sound as observed by Eegnault, is 

^- = 1.3945, 

c ' 

from which we derive, for the specific heat of constant volume 

c = .17034. 

Inserting these numerical values in (14), and putting j»o = 10333, 
we find, 

(15) . . . ■ E = 436. 

, This, then, is the value of the mechanical equivalent of heat, as de- 
termined from the best determined constants which are now obtainable. 
It is the value which we shall adopt. We must not lose sight of the 
fact, however, that there is some uncertainty as to its exact value, 
in consequence of the large result that small variations in the value of 

the ratio would produce. 

8. Adiahatic transformations. — Up to this point we have considered 
only those transformations in the state of a gas which are caused by 


its losing a quantity of heat. It may, however, happen that the trans- 
formation takes place within an envelope which is impermeable to heat j 
in which case the gas may change its volume and pressure, and conse- 
quently its temperature, without gain or loss of heat. The change is 
then said to be adiabatic, and the laws which govern it may be derived 
from equations (6), (8), and (9), by putting dq = 0. We thus obtain the 
(16) e' p d V -\- c V dp = 0, 

(17) €dT+^^~^pdv=0, • 

(18) €'dT-^'^^^vdp = 0, 


from which follow several important results. 


9. Law of pressures. — Taking equation (16), and putting n = -,itmay 


be written 

V p 

whence, integrating, and calling/ a constant, 

'/ilog r + logi? = log/, 
(19) , . v^p=f. 

Also ^, the volume of the uuit of weiglit, is the reciprocal of /?, the 
weight of the unit of volume. Thus, equation (19) may be written 
p =j:f p"^'^ or, in an adiabatic change, the pressure varies proportionally 
to a power of the density equal to the ratio of the two specific heats. 

Such is the law of pressures discovered by Laplace and Poisson be- 
fore the birth of the mechanical theory of heat. 

10. Lmv of temperatures. — Let us consider equation (17). If we put 

c' = n c, and for j> its value ^, derived from (5), it becomes, 

whence, b^^ integratiou, and calling /i a constant, 

(20) • . . Tr^-^=-/i. 

Consequently, recollecting that, as before, v is the reciprocal of the 
density, we have this law : The absolute temperature of a gas, in any 
adiabatic transformation, is proportional to a power of the density equal to 
the ratio of the tivo specijic heats minus one. 

We may also present this law uuder another form. Equation (17) 
may be written 

E cd T + ^; (iv= 0, 

(>/ ^. \ 

by recollecting that we iiave, from (13), __- = Consequently, inte- 

i\ li, 

grating, and calling To and T^, the initial and final temperatures of the 

gas, we have 

(21) Ec(To-TO= C pdv. 

But the second member represents the total work done by the elastic 
force of the gas (^o. 5). Thus, in an adiabatic transformation, the tem- 
perature is lowered by a quantity ivhich is proportional to the external icork 

11. The u'orl- done hy the indefinite adiabatic expansion of gas. — If we 
suppose that the gas expands indefinitely without gain or loss of heat, 
the volume, i\ increases indefinitely : and it results, from equation (20), 
that the temperature, T, approaches zero. The final temperature of the 
transformation Ti is then zero. Putting Ti=0 in equation (21), we have, 
for the total work of the expansion, 

(22) ^ = E c To. 

The work is therefore equal to the product of the mechanical equiva- 
lent of heat, the specific heat under constant volume, and the initial ab- 
solute timperature. 

We may thus say that the measure of this work is found by multi- 
plying the mechanical equivalent of heat by the quantity of heat which 
the gas absorbs, under constant volume, when its temperature is raised 
from the absolute zero to To : or gives out. under constant volume, when 
its temperature is lowered from To to absolute zero. 

We shall finish this exposition by a rapid summary of the laws which 
control the vohimes and specific heat.s of gases and vapors, and which 
permit us, in many cases, to calculate these important elements without 
its becoming necessary to liave recourse to experimentation. 

12. Laics governing mixtures of gases. — 1st. The specific volume of a 
mixture of gases is the compound mean of the specific volumes of the 
gases mixed. Thus, let ai, a-i, a^, be the quantities of the various gases 
which make up a unit of weight of the mixture, and V], I's, v^, the specific 
volumes correspondiog : the specific volume of the mixture is 

(23) i'0=«'l i"l+^<2<"2+ ^J3^'3+ • . . 

Example. — Taking for units the kilogram and meter, the specific 
volumes of oxygen and nitrogen are .69911 and .79607. And, according 
to the determination of Dumas and Bonssin2ault, these sases exist in 


^tmosplieric air in the ratio of .23 to .77. Substituting these values 
in (23), we have, for the specific volume of atmospheric air, i;o=. 77385. 
Eegnault found by direct experiment, -^0= -77318, which differs little 
from the first. 

2d. The specific heat (under constant pressure or constant volume) of 
a mixture of gases is the compound mean of the specific heats of the 
gases mixed. This law gives rise to the formula, 

(24) ....... c=a:Ci+a2e.2+a3C3-^ . . . 

similar to the foregoing. 

If we apply this to the heat of atmospheric air under constant pres- 
sure using the values &i ~ .2175 and c\ = .2438 for the specific heats 
of oxygen and nitrogen, we find & = .2378. Eegnault's experimental 
determination was .2374. 

13. Relation between the specific volumes and molecular weights, — Gay- 
Lussac was the first to formulate the following important law, which 
subsequent experimentation has fully confirmed : 

The product of the specific volu7ne of a gaseous hody hy its chemical 
equivalent^ or by a multiple or simple submultiple of this equivalent, is 

Thus, if we designate by Vo and e the specific volume and the equiva- 
lent, we have 

(25) m e Vo = h, 

m being a number which is generally unity, sometimes 2 or J, and very 
rarely J. 

Taking m = 1 for hydrogen, we have: 
m = 1 for nitrogen, chlorine, bromine, iodine, and all the metals which 
ha\'e as yet been volatilized. To this group belong also 
numerous binary compounds ; protoxide of nitrogen, carbonic 
oxide, carbonic di oxide, vapor of water, sulphurous acid, 
hydrogen sulphide, &g. 
m = 2 for some simple bodies : Oxygen, sulphur, selenium, tellurium, 

phosphorus, arsenic . . . 
m = J for binoxide of nitrogen, chlorhydric acid, ammonia . . . 
m = J for some very few substances, chlorhydrate of ammonia, for ex- 
If we take unity as the equivalent of hydrogen, and take the kilo- 
gram and meter as units, the numerical value of h is 11.15, about j 
so that the specific volume of a gas whose equivalent is e is given by 
the formula, 

(26) Vo = 

Gay-Lussac's law may be very simply presented, if we admit, follow- 
ing the atomic theory, that the number of molecules in unit of volume, 



at a fixed pressure and temperature, is the same for all gases. In fact,, 
according to this hypothesis, the weight of unit of volume is propor- 
tional to the weight of a molecule, or to the molecular weight. But the 
weight of unit of volume, and the specific volume Vo are inversely pro- 
portional; hence, from (26), the molecular weight is proportional 
to 7)1 e. 

Thus, tbe multiple or submultiple of the equivalent which satisfies 
the law of the specific volumes is a measure of the molecular weight. 
For example, the molecular weight of hydrogen and nitrogen being H 
and iS", those of oxygen and suli)hur are O^ and S^. The molecular 
weight of ammonia and chlorhydrate of ammonia, for which the values 
of m are i and i, are :N'^ H^- and X* H Cl< 

We may thus formulate as follows the law of specific volumes : The 
product of the specific volume of a gas by its molecular iceight is constant. 

14. Laics of specific heats. — Molecular heat, — The product of the mole- 
cular weight of a body by its specific heat is called the molecular heat 
of the body. 

Regarding gases, we have from experiment : 

1st. That the molecular heat under constant pressure has the same 
value, K', for hydrogen, nitrogen, and oxygen. 

2d. That it is greater than K- for the other gases and vapors, and 
that it becomes greater as the gas becomes more complex. 

The numerical value of K', derived by taking the mean of the specific 
heats of hydrogen and nitrogen, may be taken as 3.411. 

If, then, we call c' the specific heat under constant i)ressure and (o the 
molecular weight, we may write — 

(27) 0' = "^^+^ 

a being a quantity which is equal to zero for the simple gases, and those 
the nearest to the state of a perfect gas; and is positive for others, and 
increases as the molecule of the gas or vapor is made up of a larger 
number of atomic elements. 

For example, for vapor of water, carbonic sulphide, alcohol, and 
ether, whose molecular weights are 


HO C S^ C^ W O C^ W O, 

the values of a are respectively 

.913 2.555 7.017 14.386. 
15. The specific heat under constant volume, whose experimental de- 
termination is almost impossible, is calculated for perfect gases by the 
relation (14), which gives 

<2«' • '-'-m'-^'- 


It is only necessary to know e' and the specific volume ^o- The 
latter may be determined by experiment, or deduced from formula 
(26), when the molecular weight w is known. We have, in this case, 

^0 = _ ; and consequently putting 

(29) ^ = ^'-2^3^' 

and replacing c' by its value (27) in equation (28), the latter may be 

(30) c = ^ + ''- 


The value of K, the molecular heat under constant volume of simple 
perfect gases, is obtained from numerical values of quantities on which 
(29) depends. 
. We have already shown that 

K'=3.411 /t=11.15 E = 436 i?o=10333, 

whence it results K =2.443. 

16. Relation between the two specific heats. — This ratio, which is so im- 
portant a factor in all adiabatic transformations, is deduced from (27) 
and (30) ; which give 

<»« ........ -M^t.;. 

For hydrogen, nitrogen, and oxygen we have, very nearly, a = ^ 


K^ 3.411 -, ,^ T 

For gases and vapors whose molecules are complex, the value of n 
approaches rapidly to unity for increasing values of a. 

If we admit, for example, the values of a already given {Iso. 14), we 
have, for vapor of water, sulphur, carbon, alcohol, and ether : 

71=1.29 1.19 1.10 1.06. 

17. Finally, the value ?i=1.40, which we have found for those of the 
simple gases which approach most nearly the ideal state, which has been 
called the state of a perfect gas, appears itself inferior to the theoreti- 
cal limit which would be realized if the molecules of the gas were 
strictly mono-atomic. It results, in fact, from reasoning based on the 
new thermodynamic theory, that the ratio of the two specific heats has, 

in this case, the extreme value n = -. The smaller value (1.40) found is 




explained by Gehrardt aud Clausius by the liyi)c)thesis that the mole- 
cules of even these gases contain at least U\o atoms. 

We are thus led to conclude that the ratio of the two specific heats, 
variable with the greater or less complexity of the body, is between the 

limits - and 1 : and approaches rapidly to the inferior limit when the 

number of atoms making up the molecule is increased. We shall see 

hereafter that these considerations, which may appear at first purely 

speculative, are not without importance for the particular object in view. 

18. Hypothetical laic of Claiishis. — It is known fi^om the experiments 
of Eegnault upon the specific heats of solids that: 

1st. The molecular heat of simple bodies is constant iDulong and 
Petit's law). 

2d. The specific heat of compound bodies is the compound mean of 
the simi:)le bodies of which they are composed. 

Clausius supx)oses that these two laws may be applied to the specific 
beats under constant volume of perfect gases. According to this hy- 
pothesis the specific heat under constant volume of a perfect gas may 
be found by the relation, 

(32) c= ^ 

UJ , 

o> being its molecular weight and K the constant, whose definition and 
value have already been given (^o. 15). 

The specific heat of a comi^ound gas is deduced, by formula (21), from 
its chemical composition, as if the combined elements were mixed. If, for 
example, a gas whose composition is represented by the formula 

n\ n-i nz 

wi, ^2, C/J3 being the molecular weights of the constituents, so that 

at = )ii cui -{- n., oj.-, -\- n^ ^j 

will be the molecular weight of the compound. We apply here formula 
(24), remarking that the quantities of the elements in unit of weight of 
the compound are 

Consequently, calling Ci, Og, C3, the specific heats under constant volume 
of the constituents, and c that of the compound, we have 

C — ^L_*^ L^i + Tig a>g Cg 4- ^3 w^ C 3 

and finally, from the relation (32), 

(33) ....... c="^ ^^'^ ■^^"- "^--^^^^- 

Pattipg Wi H- ??2 + ^3=^, and piittiDg Vo=— and c= in equation 

(28), and from (29), we have, 
(34) ........ c'= ^' + (-^-1) ^ . 

From (33) and (34) we have, finally, 

(^^) »=4=^+Ke-0' 

which, from the value ^ = 1.40, already found (]^o, 16), for perfect sim- 
ple gases, may be written in the very simple form, 

^''^ • • • » = i + 54- 

20. It is hardly necessary to dwell upon the importance of the pre- 
ceding formulas. They show that the specific heats of gases and 
vapors depend upon their cheaiical composition. 

The application of these formulas, even to perfect gases, must how- 
ever be under certain reservations. For if, as shown by the experi- 
ments of Regnault, hydrogen, nitrogen, and oxygen satisfy sufficiently 
accurately (32), it is not so for chlorine and bromine; and the departures 
from the law are sufficiently great to make us attribute them to some- 
thing else than alterations in the laws of Mariotte and Gay-Lussac. 

Also, formula (33) gives greater values to the specific heats of com- 
pound gases than those found by exijeriment. In this last case, how- 
ever, the error must be lessened at high temperatures ; for the specific 
heats under constant pressure of imperfect gases increases with the 
temperature, as Regnault has shown by experiment upon carbonic 

. If it is not, then, certain that the hypothesis of Olausius gives rigor- 
ous formulas in the limiting state, we can at least admit that it fur- 
nishes an approximation equally good with that furnished by the 
experimental laws which govern the specific heats of simple and com- 
pound solids ; and that the relations which are derived from it, so fre- 
quently used by Bertholet in his researches in thermochemistry, may- 
be reasonably accepted. 

We shall apply them in the following chapter to the theoretical cal- 
culation of the principal elements which are required for the approxi- 
mate evaluation of the force of explosive substances. 



1. When a substance is exploded in a capacity of constant volume 
the products of combustion develop a pressure whose value, variable 
with the time, reaches a maximum in an extremely short space of time, 
and then decreases progressively in consequence of cooling. At the 
end of this cooling the substance is changed into products whosenatura 
is variable with the conditions of the exiDlosiou, and which may be de- 
termined by chemical analysis. 

These products are, according to circumstances: 

Exclusively gaseous (detonation of chloride of nitrogen) ; gaseous 
and solid (explosives of the nitrate and chlorate class) : gaseous and 
hquid (nitro-glycerine and gun-cotton). 

In all cases we must admit that, at the instant of maximum tension, 
the loroducts are totally gasified ; the temperature at this instant is 
generally greater than the melting point of the most refractory metals. 

Also, at this temperature, the i^roducts of combustion are probably 
not the same as those found after the cooling. As remarked long sinc& 
by Melsens, they are then i^artially or totally dissociated; and, by a 
chain of transformations producing gradually less simple combinations, 
the variable state of temperature and pressure brings about the fiual 
state, which alone can be the subject of observation. 

2. Force of an explosive substance. — If we suppose that the laws of 
Mariotte and Gay-Lussac are applicable to the gasified products of ex- 
plosion at the instant of maximum tension, the value of this tension, 
as a function of the temperature and specific volume corresponding, is 
easily obtained. 

Suppose, for example, that we consider unit of weight of an explo- 
sive detonating in volume i\ The maximum pressure will be given by^ 
the equation 

(37) ....... p'--=^^^. 

in which To is the absolute temperature of the products of explosion at^ 

the instant of maximum tension : Vq the specific volume of the mixture 

of these products, supposed entirely gasified. 

Consequently, we have the pressure, which is inversely proportional 

o V T 
to the volume, equal to J-^^A_J, when the volume is reduced to unity. 

^ to 


Tbe quautity 

(38) /=i'»JL^» 

is what we shall call the force of the explosive. It represents the pres- 
sure developed by the unit of tveight of the substance detonating ifi unit of 

We shall next see how its approximate value may be determined. 

3. Heat and temperature of combustion. — We shall call heat of com- 
bustion the quantity of heat Q that unit of weight of the explosive sub- 
stance evolves, under constant volume, in the ideal case where the final 
temperature of the products of combustion is absolute zero. 

The absolute temperature of combustion is the terai)erature T, which the 
products would have if all the heat of the combustion were used to heat 
them, from absolute zero. 

If the specific heat of the products of combustion under constant vol- 
ume were constant throughout the transformations which these prod- 
ucts undergo in passing from absolute zero to T, or, inversely, from T 
to absolute zero, we would have between Q and T the well-known rela- 

(39) T = ^. 


The specific heat, c, would be practically constant if, in the changes 
undergone, the products remained gaseous, and if, in this state, the hy- 
pothesis of Clausius were really apijlicable to them. In fact the spe- 
cific heat of acompound gas being, according to this hypothesis, the same 
as if the combined elements were mixed, it is clear that the mean spe- 
cific heat of the mixture preserves, in all states, a value equal to that of 
the entirety of the simple bodies which make up the explosive body, 
supposed freely mixed. 

In reality, there is reason to believe (Oha])ter I, No. 14) that this is 
only approximate, and that the specific heat increases in transformatious 
in which the constituents pass from a less to a greater state of complex- 
ity. It also certainly increases at the time of the passage of a body 
from the gaseous to the liquid state, and from the liquid to the solid 

Consequently, equation (39) would give too great a value for T, if we 
take for c a value derived by the hypothesis of Clausius for the entirety 
•of the products supposed gaseous. We would have too small a value 
•on the other hand if we follow Bunsen and Schisckhofi", and take the 
value relative to the final state. 

We shall adopt the first value, which, as Berthelot has remarked, 
seems to be preferable for reactions whose temperatures are high, and 
which, moreover, has the advantage of enabling us to calculate the value 


<i priori from tbe composition of the explosive substance and inde- 
pendently of an analysis of the products of explosion. 

4:. We shall suppose also, following all the authors who have treated 
subjects analogous to the one we have in view, that the temperature, Tq, 
corresponding to the maximum pressure, is sensibly equal to the tem- 
perature ot combustion, T, already defined, ^'e write, consequently 

(10) T, = ?. 

Thus we have a superior limit: for the heat of .combustion. Q, com- 
prises not only the quantities of heat which change the temperature of 
tbe products, but also those which the chemical or physical changes of 
state i^roduce. 

These principles being admitted, we obtain the definite expression of 
the force of an explosive substance in this form. 

noh) f=^ . ^\IiR. 

- > d c 

The force then is directh/ proportional to the heat of combustion and the 
speciiic rolumej and inrersely proportional to the specific heat. 

It remains now to show how these three characteristic elements may 
be theoretically or experimentally determined. 

5. Value of the specific heat under constant volume. — According to the 
hypothesis which has been admitted, the value of c is obtained by apply- 
ing the law of gaseous mixtures (Chai)ter I. Xo. 12 1 to the elements 
which make up the substance under consideration. 

If. for example, ai, Og, ag, are the proportions of the various elements, 
and u)^, (v^. u>3. their molecular weights, for nnit of weight, the correspond- 
ing specific heats are, according to equation (32): 

K K K 

and we have, consequently, according to formula (24 . for the si^ecific 
heirt sought, 

(41, .... ,: = k(^ + ^+^+ .. \ 

\^ Oil W., 03 y 

This formula requires that the molecular weight of the elements shall 
be known. The following are the values for those simple bodies which 
make up the most common explosives. 
5781 2 


Name of body. 

Moleculai weight. 

Symbol. Value 

Nitrogen ., 
Chlorine. . . 
Sulphur . . , 
Carbon ... 
Sodium . - . 









6. Another method may be given for the calculation of c. Suppose, 
first, that the substance is a definite compound, like nitro-glycerine or 
gun-cotton 5 and that its equivalent e is represented by the formula 

ni n2 ns 

wi w^ o}^', Wj, ^2? <*^3? being the molecular weights of the elements, so 
that we have 

e*=^ 111 w^ 4" ^2 "^2 + '^3 ^3 + 
The proportions of the elements in unit of weight being 

__ '^h ^1 

^2 ^2 

«3 =- 


formula (41) becomes 


_ K (^1 + % + %+'•') 

Notice that, in this expression, as well as in the equivalent one (41), 
the numerical value of K is (Chapter I, No. 15), 

K = 2.443. 

Let us take, for example, chloride of nitrogen, N OP; we have 

^ = 14 + 3 X 35.5 = 120.5, n, = 1, n, = 3, c = ^^^~ = .0811. 
Again, let nitro-glycerine be represented by the formula 

06 H^ (N O' Rf = 0« H^ W Oi« ; 

in this case, e = 227. 

Writing the formula in the form 

in order to show the molecular weights, we have 
_ 2.443 (3+5+3+9) _ ^197 

_ 227 

7. When the substance is a mixture we may first calculate the specific 

' 19 

lieat for each of the constituents, and then take the compound mean 
according to the proportions of the ingredients. 

The following table shows the values of c for the simple and com- 
pound bodies which compose the principal explosives : 

Name of substance. 



Nitrate of potassium . 

Nitrate of sodium 

Chlorate of potassium 
Cliloride of nitrogen . . 

Nitro-glycerine I €« H^ (N 0^ H)3 


Picrate of potassinm 


Value of c. 






N 05 Na 


CI 05 K 




C6 H2 (N 06 H)3 




Ci2 H2 K (N 04)» 02 


The following are the values of c for the various powders made in 
France : 


Kind of powder. 

Best sporting powder 

Cannon powder 

Fine-grained jtOT?der, called 1 B. 

Powder of commerce 

Ordinary blasting powder 


of c. 











These values differ very little, and may all be taken as equivalent to 
their mean, .1445. 

It has also been found that for a mixture of 55 parts of picrate of 
potassium and 45 of saltpeter, c = .1661; and, for a mixture of equal 
weights of x>icrate and chlorate of potassium, c = .1513. 

Note. — In the calculations relative to gunpowder the charcoal has 
been treated as pure carbon. The charcoal used in the manufacture of 
gunpowder contains, however, hydrogen and oxygen about in the pro- 
portion in which they exist in water. This will cause an increase in the 
value of c, and one which may become appreciable, since the value of c 
for vapor of water is .4072. 

9. Theoretical determination of the heat of combustion. — The heat of 
combustion, which, like the specific heat, is one of the characteristic ele- 
ments of the force of explosive substances, may be calculated when we 
know : 

1st. The chemical composition of the substance and the final products. 

2d. The quantities of heat absorbed or given out by the formation of 
the compounds which make up the initial and final state of the reac- 


It results, in fact, from one of the fiindameutal laws of thermo-cliem- 
istry, that the amount of heat given out by a continuous series of reac- 
tions is equal to the difference between the heat of formation of the 
compounds in the initial and final states. 

Berthelot, w^ho was the first to state this law, basing it upon the well- 
known law of the conservation of energy, has made numerous and im- 
portant applications of it to the heats of combustion of explosive sub- 
stauces. It is thus made possible, by studying the calorimetric researches 
of various experimenters, and, without making further determinations, 
to explain in a rational manner the characteristic properties of the det- 
onating substances, which, before the utterance of this law, could be 
compared only by eminrical methods. 

We shall not dwell further upon these interesting researches, but, re- 
ferring the reader to the works of that eminent chemist for further de- 
tails, shall reca]l some experimental laws which bear upon the same 

9. Experimental determination of the heat of comhustion. — To determine 
by experiment the heat of combustion of an explosive substance we 
have only to fire a known weight of it in a vessel plunged in a calorim- 
eter, and to observe the increase of the. temperature of the bath when 
it has absorbed the heat of the reaction. 

The first determination of this nature was made by Buusen and 
Schischkoff' upon a powder like our sporting powder. The experiment 
showed that the ])roductsof combustion of a kilogram of this i)owder 
give out GI4 calories in cooling to a temperature of 20 degrees. 

We have attempted, in a series of experiments at the central depot 
of the State manufactory, to resume these researches, and to extend 
them to various substances. We shall give, further on, the results of 
these experiments : 

Calorimetric determinations do not give all ihoi heat of combustion, 
Q. The measured quantity, g, is that which is given out by the prod- 
ucts of combustion in passing from the temperature of combustion, T, 
to the absolute temperature of the calorimeter t. The difference, Q— g^, 
represents, then, the heat which would be given up in passing from t 
to absolute zero. It is generally very small, compared to Q, because of 
the extreme elevation of the temperature of combustion. W^e may, more- 
over, easily find its value if we knew the specific heat, c, between the 
limits of t and zero. We would have, 
(43) ...... Q = ^ + ot 

The specific heats corresponding to the lower temperatures are, in 
general, not known ; for the want of more precise data, we may substi- 
tute for them the theoretical values found in iTo. 7. These values are 
too small ; but the resulting error, affecting only a relatively very small 
term, would be small. The mean temperature of our experiments being 
17 degrees, the corresponding value of the absolute temperatures is : 
^ = 273 4-17 = 290. 


We may thus, for these elements, aiDproximately correct the observed 
heats, and derive from them the correspouding heats of combiistioD. 
The following table gives, for the principal exi^losives : 
1st. The quantity of heat </, measured by the calorimeter ; 
2d. The heats of combustion Q, deduced from the foregoing by (43) 5 
3d. The temperatures of combustion, calculated from the heats of 

combustion Q, and the theoretical specific heats by the formula T 


Heat of combustion 

Name of esplosiTe. 

Measured Corrected. 
q. \ Q. 

of combus- 



B est sporting powder 

Cannon powder 

Fine-grained powder, called B 

Commercial powder 

Ordinary blasting powder 

Chloride of nitrogen 



Picrate of potassium 

Mixture of 55 picrate of potassium and 45 saltpeter . 
Equal parts of picrate and chlorate of potassium... 

807 ! 
553 I 
1,180 i 












Degrees cent: 
4, 590 
■ 8, 09O 

Tliese figures are the results of calorimetric experiments made at the 
central depot, except those for the heat of combustion of chloride of 
nitrogen, which were made by Deville .and Hautefenille. 

10. Theoretical determination of the specific volume Vo- — The specific 
volume of the products of combustion ijresents greater difficulty in its 
determination than any otlier of the elements. For we can only ob- 
serve the products in their final state, and can know nothing experi- 
raentaily of their condition at the instant of maximum tension. We 
can, however, make a calculation according to two hypotheses, corre- 
sponding to two extreme cases which comprise, probably, the truth. 

We can make the calculation as though the elements of the mixture 
were entirely dissociated, having thus a superior limit of the possible 
specific volumes; and we have also an inferior limit by supposing that 
the compounds found in the final state are already formed, and are 
simply vai)orized at the time of maximnm tension. 

First case. Complete dissociation. — Resuming the notation of ISTo. 5, 
let us call a'l, <a'2, a'2, . . . the proportions of the elem.ents forming 
unit of weight of the substance, and gji^ oj-,, cja, . . . their molecu 
lar weights. The corresponding specific volumes are (chap. I, Ko. 13) 

h h h 

Vi = , 1-2= , y-, = , 

and the specific volume of the mixture is, by (23), 
(44) . . . .,„ = ,Y^ + ^ + ^+ . ] .Y 


Second case. Vaporization of the final products. — If the final products 
have been found by chemical analysis, the specific volume of each one 
may be calculated from its molecular weight, which, generally, is equal 
to its equivalent. This computation involves no other possible error 
than the one which results from our being obliged, in certain cases, to 
take the half of the equivalent. The cases in which the molecular 
weight is one-fourth of the equivalent are very exceptional, and may be 
left out of account. 

After the preliminary calculation, it is only necessary to take the 
mean of the specific volumes of the bodies forming the final state ac- 
cording to the proportions of the mixture. 

11. The calculation is simplified by applying first the equivalent of 
the explosive substance, and afterwards reducing to unit weight. If 
the composition in the final state is known, in equivalents, the specific 
volumes in the two limiting cases may be simply calculated at the same 
time. Suppose, for example, that the explosive is represented by the 

formula w/" mJ^' w^^^ . . . .; wj, ^3, ^3, being the molecular weights 

of the elements, so that 

e = Ui oji -f- ^2 uj., -\- its ojr^ -\- . . . 

is the equivalent of the substance. 

Call III, II2, II3, • • • the molecular weights of the bodies of the 
final state, and 

e = Ki III, N, II2, N, II3, ... 

the formula of the reaction. 

Eeasoning as we have already done several times, particularly to ob- 
tain formula (42) for the specific heats, we easily find that the specific 
volume is represented by the formula 

/.rx ■ ^ ('^1 + n,+ ns+ » » . ) 

(45) .... v, = ^ 

in the case of entire dissociation ; and by the formula 

(46) .... .„='L(Il+Jl±ZH-_^^) 

in the case of the voJatUziation of the compounds of the final state. The 
value ofh is (chap. I, Ko. 13), h = 11.15. 

12. We consider, for example, nitro-glycerine, whose combustion is 
represented by Beithelot by the equation 

C^ H^ (NO^H)^ = 6 CO^ + 5 HO 4- 3 K + O, 
or, showing better the molecular weights, 

[Cy WW (0^)3 = 6 GO'^ + 5 HO -fr 3 N H- i O'^ 

Consequently, the specific voUinie i>. 

11.1 ,. -p ^ ., ^ J) 
I „ — 227 ~ .c»oo, 

in the case of complete dissociation : and 

^ _11.15 ((3 4- 5 4- 3 + ^) _ .,.> 
Iq — 227 — .< 1^, 

in the case of the vaporization of the bodies of the final state. 

In the following- table are written the calculated results for some ex- 
plosive bodies. The formulas of reaction adopted are those used by 
Berthelot in his researches : 

Xame of explosive. 

I Speciac 


Of elements 

Of products 




Chloride of DitTO'''t?n 

370 ! 


Nitro- <''ly c 6riD 6 






Picratft of pot;i,ssinni 



^ .— 1 

13. Intermediate states. — We see, from the table, how great an in- 
fluence the chemical state of the products at the instant of maximum 
tension have upon the value of the specific volume. The ratio of the 
two limits, which is unity for chloride of nitrogen, which always decom- 
poses into its simple elements, varies for the dilferent bodies between 
that limit and two, for the best sporting powder. 

^ow, between these two extreme cases, there may be intermediate 
states, according to the varying physical conditions of the products in 
the very short period between the beginning of the bnrning and the in- 
stant of maximum tension. It is very probable that the mode of com- 
bustion may exercise a very great influence on these circumstances ; 
and that the conditions, for example, which develop the reaction in- 
stantly in all parts of the body, difler essentially, from this point of 
view, from those which cause a progressive reaction. 

It is probable that we may ascribe to these causes the diflerent effects 
produced by the same substance under varying conditions. 

14. Theoretical calculation of the force of an explosive substance^ when 
totally dissociated. — Following our examination of the circumstances on 
which the characteristic elements of explosive substances depend, we 
may lay down some considerations regarding the theoretical expression 
for the corresponding force. 

This expression is given by formula (40^), and its value may be found 
when, by the aid of the preceding principles, the values of c, Q, and y,j are 


known. Among the results which are thus obtained, we must notice- 
that which relates to the complete dissociation of the products of com- 

In this case, according to the values of c and Vq (41) and (44), the for- 
mula reduces to 

(47) . . /-i^AQ 
^ ^ /-273 K' 

and, as the constants li and K are independent of the nature of the sub- 
stance, we are led to the following consequence : The force of an explo- 
sive substance when entirely dissociated^ is proportional to its heat of com- 

We have, also, h = 11.15, and K == 2.443 j inserting these values in 
(47), and putting j?o==l, we have for the expression of the force in at- 

(48) ........ /= .01672 xQ. 

This is evidently a limit superior to that actually reached. 

15. Intermediate states — Equivalent combustions. — There is reason to sub- 
pose, as has already been stated, that the chemical state of the products 
at the instant of maximum tension is not generally that of complete dis- 
sociation ; the actual state is less simple and varies with the conditions 
of combustion. 

Among the different reactions which these conditions may cause, we 
note those which iDresent the same initial and the same fiual states. To 
these reactions the following principle, laid down by Berthelot, in his 
thermochemical researches, under the name of the '' principle of the 
calorific equivalence of chemical transformations," apply: 

Having given a system of simple or compound bodies in a determined state; if this 
system undergoes any physical clianges bringing it to a second state, the quantity 
of heat absorbed or given out depends solely upon the initial and final states of th© 
system. It is the same, whatever may be the nature or sequence of the intermediate 

Let us call, for brevity, those reactions which have the same initial 
and final states, calorifically equivalent,, or simply equivalent reactions. 

Observing that : 

1st. Equivalent reactions have the same heat of combustion Q ) 

2d. The specific heats of differing equivalent reactions is the same, 
since, according to the hypothesis of No. 3, it depends only upon the 
nature and proportions of the simple elements present in the reactions^ 

Thus the force varies only as the specific volume Vq) and we have 
this consequence : 

The force developed by the same explosive substance^ in varying equivalent 
reactions or combustions,, is proportional to the specific volume of the prod- 
ucts formed at the instant of maximum tension. 

Bv the different values of this element, variable with the state of dis- 


sociation of the products, may be explained the considerable variations 
of the force ; even though chemical analysis of the final products and 
calorimetric determinations remain the same. 

16. Theoretical calculation of the force of the final products ivhen vapor- 
ized. — The lower limit of the force of an explosive is obtained by the aid 
of the value of i?o, corresponding to the final state of the products, sup- 
posed entirely gasified, but not decomposed, at the instant of maximum 

Making the calculation for the substances whose specific volume has 
already been found (No. 12), and comparing them with those which are 
obtained by the formula for total dissociation, we have the following 
table : 

Name of explosive. 


Spoi ting powder 

Chloride of nitrogen 
Nitro-glycerine . . : . . 


Picrate of potassium 






Final state 

12. 08 
11. 25 

The linear unit adopted being the meter, the table gives, in atmos- 
pheres, the pressure developed by a kilogram of the substance deto- 
nating in a cubic meter. 

17. Approximate expression for the force of an explosive substance. — We 
may obtain another expression for the force of an explosive substance- 
It is only approximate, but is extremely simple, and has the advantage 
of containing only quantities which may be determined by direct exper- 
iment, without its being necessary to have recourse to the determina- 
tion by chemical analysis ; which, as the foregoing considerations show?- 
depend upon the theoretical and somewhat unreliable values of the 
specific volumes. 

Take the general expression for the force, 

•^ 273 c ' 

From equation (14), which is between the heat and specific volume of 
a gas, we find 

1 £o^o _ 
273 c 


Consequently, calling?! the ratio — of the two specific heats, we have 


the very simple formula, 

(49) . /=(^ — l) EQ. 


The ratio n is equal to 1.40 about for hydrogen, nitrogen, oxygen, car- 
l)onic oxide, &c. We have, consequently, 


/=|e q, 

if these gases, or others approaching nearly the state of a perfect gas, 
were the only products of combustion. But, in a great many cases, 
these products comprise besides vapor of water and other even more 
complex vapors, which are liquefied or solidified after the cooling. For 
these compounds, 7i is less than 1.40, and approaches unity, in conse- 
quence of the progressive formation of the final state. 

If, then, we call s the weight of i3ermanent gas due to the combustion 
of a kilogram of the explosive substance, the quantity n — 1 may, in 
a first api^roximation, be considered as a function of s, which, vanish- 

ing with e, becomes ^forf=l. 


We have then, nearly, n — 1 = - s ; and, 


E Qf. 

This formula shows that, similarlj^ to a law of Berthelot, the force of 
explosive substances is nearly proportional to the product of the heat 
of combustion by the weight of the i)ermanent gases produced by the 

If the conditions of the explosion are such that the substance is en- 
tirely, or even partially, dissociated, so as to i)roduce compound gases 
for which we would have ?i=1.40, as would be the case for carbonic 
oxide, we must take £=1, and the force would be given by (50). The 
latter agrees, as may easily be shown, with the value (47) already ob- 

18. By means of experiments made at the central depot, € has been 
determined for some substances. These, together with the heats of com- 
bustion, and the corresponding forces, exx)ressed in atmospheres, are 
tabulated below: 

Name of explosiv^e. 

"Sporting powder 

Common powder 

Fine-grained powder, called B 

Powder of commerce 

Ordinary blasting powder 

Chloride of nitrogen 



Picrate of potassium 

55 parts of picrate of potassium and 45 of saltpeter 
Equal parts of picrate and chlorate of potassium . . . 

of gas. 



Heat of 








1, 123 

■ 840 




I tm. 




5. 15 






19. Eelative force of the explosive substances. — It is remarkable that 
the five powders should have about the same force, notwithstanding 
differences of fabrication. This result has been contirmed by experi- 
ment, which shows that the bursting charges of the five different 
powders for the same shell varies only 15 to 17 grams. 

The mean of the forces of powders is 5.29. This corresponds to a 
force of 5,290 atmospheres for a kilogram of powder, detonating in a 
liter, that is to say, in its own volume ; these results are thus in accord 
with those made by Captain l^Toble upon i>owder exploded in its own 
volume. The pressure measured by him under these conditions by 
means of a gauge, was 37 tons on the square inch, or 5,000 atmospheres, 
for the F.G. powder, and about 32 tons for the E.L. G. The following 
table shows the relative force of the different explosives : 

JName ot explosive. force. 

Powder containing saltpetei' 1- 

Chloride of nitrogen 1-08 

Nitro-glycerine i- 55 

G-uu-cotton 3. 06 

Picrate of potassium 1. 98 

55 parts picrate of potassium and 45 of saltpeter ' 1. 49 

Equal parts picrate and chlorate of potassium 1. 82 

20. The figures of this table appear to agree sufficiently well with 
the real effects of explosives, fired in a manner to produce what, in the 
experiments undertaken with M. Eoux, we have called explosions of 
the second or<?er, that is to say, explosions produced by any other agency 
than a strong fulminating primer. 

For example, the relative force of gun-cotton is about equal to that 
found by the French commission on gun-cotton (3.20), derived from the 
comparison of the charges just necessary to burst a shell. 

Other similar experiments confirm the result for nitro-glycerine. The 
mean charge of powder required to rupture shell was 16 grams. In 
the same shells, the effect of dynamite which contained 50 per cent, of 
nitroglycerine, fired by a small quantity of powder, was such that 1 
part of nitroglycerine was equal to 2 parts of ]3owder, under these cir- 
cumstances. But, as Berthelot has remarked, the heat disengaged by 
firing dynamite is divided between the products of the explosion of the 
Tiitro-glycerine and the inert vehicle, which latter has about the same 
specific heat. It results that, in such dynamite, the temperature, and 
consequently the force, must be lowered one-half. The force of j3ure 
nitro-glycerine should then be doubled, and be estimated at about 4. 
The difference between this value and that given in the table is not 
great. We will add that, for a mixture of picrate and nitrate of potas- 
sium, the bursting charge was found to be about 11 grams, which 
corresponds to a relative force of j f = 1.15 ; which differs little from the 
theoretical value, 1.19. 


21. On the relative force of some substances^ ivlten dissociated. — lu ap- 
plying formula (51) to nitro-glycerine and gun-cotton, we suppose that 
those of the products which are not permanent gases after the cooling, 
are vai)orized, but not dissociated, at the instant of maximum tension. 
If they are, on the other hand, decomposed into gas so that the value 
of 71 is 1.40, we must use formula (50). In comparing, as before, the new 
values with the mean force of powder, we obtain the following relative 
forces : 

Nitro-glycerine 5.68 

Gun-cotton 3. 58 

In these new conditions, then, the force of gun-cotton becomes almost 
four times that of powder. 

As to the figure for nitroglycerine, it appears to have been almost 
reached in an experiment made by M. Eoux, where about 2.7 grams of 
pure nitro-glycerine were fired by 1 gram of i)owder inan ordinary shell. 
In this experiment, 2.7 grams of nitroglycerine equalled 15 grams of 
powder; whence the value of the relativ^e force is 5.55, which differs 
little from its theoretical value, 5.68. 

22. Explosions of the first order. — But, however great the power of 
dissociated nitro-glycerine may be, it still fails to account for the effects 
observed when this substance is tired by violent percussion or by a charge 
of fulminate of mercury. It results, in fact, from our experiments, that 
the effect of nitro glycerine is then at least nine times that of powder. 

We have proved, by a new calorimetric experiment, that the heat dis- 
engaged under these circumstances does not differ sensibly from that 
due to explosions of the second order. Though this proof, made upon 
a small quantity of dynamite, on account of the intensity of the effects 
produced, has iiot the precision of our former determinations; it excludes 
the possibility of ascribing the great variation in the effects produced 
to variations in the heats of combustion. We may add that^ formula (50) 
being applicable to the complete dissociation of the substance, we can- 
not supi)ose that the explosion of the first order can produce new de- 
compositions into simpler elements. 

. We must, then, either give up the attempt to explain the phenomena 
by Mariotte's and Gay-Lussac's laws, of which formula (49) is a rigid 
deduction ; or suppose that ??, the ratio of the two specific heats, acquires, 
at the instant of maximum tension, a greater value than 1.40 ; this lat- 
ter value is that corresponding, in normal conditions, to simple gases, 
near the state of a perfect gas. If, adopting this view, we seek to de- 
termine a value of n which will make the force of nitro-glycerine nine 
times that of powder, according to the equation, 

(?i - 1) E Q = 9 X 5.29 X 10333, 

we find, taking for Q the value of nitro-glycerine, 1784, 

ri = 1.632. 


This value of w, almost equal to |, reaches^ then, the upper limit found 
theoretically for mono-atomic gases (chapter I, ^o. 17). 

By pushing further the deductions from this result, we would be led 
to explain the observed effect by supposing that ; under the influence 
of a \iolent commotion there may be produced, in certain substances, 
an unstable molecular structure, a sort of intramolecular dissociation, 
capable of destroying, during an infinitely short space of time, the 
atomic equilibrium, and of causing separation into more simple ele- 
ments than those which are produced by the normal mechanism of the 
internal forces. 

\ye do not disguise the hypothetical nature of these theories ; we have 
thought it better to formulate them, in the absence of better explana- 
tion, in order to show that these obscure phenomena are not absolutely 
inexplicable in the present state of science. 

23. If our explanation is correct, and if other unstable nitrogenous 
CDmpounds, such as chloride of nitrogen and gun-cotton, can experience, 
like nitro glycerine, this atomic dissociation when violently detonated, 
the corresponding force will be obtained by putting n = |^ in formula 
(50). It is then represented by the expression 

(52) / = f EQ, 

anrl its value is, consequently, f of that which is produced in the case 
of molecular dissociation. 

The relative forces calculated by this formula are 

Chloride of nitrogeu 1. 80 

Nirro-glyceriiii- 9. 49 

Gnn-cotton 5. 97 

Under these conditions, then, chloride of nitrogen wiU produce double 

the effect of gunpowder : and this result is certainly not at variance 
with what we know of the violence of this substance. 

24. The foregoing considerations seem likely to throw some light upon 
the cause of the varying: effects produced by explosives according to the 
m -inner of their inflammation. They constrain ns to admit, for each sub- 
stance, the existence of a sort of scale of pressures ; the point of which 
reached by the explosive, when the circumstances of the combustion 
are progressively varied, corresponds to so many sudden variations of 
the law of tensions. 

Nitroglycerine offers a striking example of this. Its explosive force, 
reduced in the x»roportion of 5 to 3, when the detonation ceases to be 
*' iulminante'Ms still more greatly lowered when the cooling of inert bodies 
6u])presses the dissociation of a part of the products of combustion. Do 
these characteristic effects of the definite explosive compounds appear 
in the same degree in the combustion of the mixtures which make up 
the ordinary gunpowders ? 

Although these substances have a relative stability, it is probable 
that they produce analogous phenomena ; and it is probably to circum- 


stances of this nature that we must ascribe the origin of the discontinu- 
ities which the discussion of Rumford's experiments has enabled us to 
state in the form of the law that the tension of the fluids of the powder 
depend on their density ', and perhaps also the variations in the de- 
termination, by different persons, of the force of powder. 

Experiment alone can solve this complex problem, whose complete 
solution will furnish the principal element of the dynamic theory of ex- 



1. We liave oousidered in the preceding chapter the pressure devel- 
oped by the combustion of an explosive substance in a fixed and resist- 
ing envelope. We shall next study the effects produced by the expansion 
of the gas in a variable volume, as is the gas in a gun. fi^om the dis- 
placement of the projectile. 

At first, we shall consider only the more simple of these effects, in 
which, the explosion being instantaneous ^ we assume that the body is 
entirely gasified before the volume has appreciably altered. The case 
of a progressive combustion, in which the gases are produced during a 
change of volume, is much more complex, and will be made the subject 
of a special study. 

2. Work due to the e\q)ansion of an explosive body. — Let unit of weight 
of the explosive deflagrate instantaneously in a capacity whose initial 
volume is i'oj call v the volume at any instant of the expansion: and 
let us find the work corresponding to the variation of volume v — r,,. 

If we neglect the loss of heat due to the cooling of the walls of the 
envelope, the transformation is adiabatic : that is, unaccompanied by 
gain or loss of heat. Consequently, the work corresponding, o,is given 
by (21), of the first chapter : and we have 

(53) ^ = Ec- Tc-T); 

c being the specific heat of constant volume of the products of com. 
bustion, and To and T the temperatures corresponding to the volumes 
Vo and r. 

u may be expressed as a function of the volume when the ratio of 
the two specific heats of the gas remain sensibly the same during the 
expansion. Equation (20) shows that Tr "-^ is then constant. 

We thus have 


T r =Tc 

and consequently 

whence we derive the new value 

(54) ^=E ^T'[l -(':)"■]• 



But accordiDg to equation (40).tlie product c To is equal to the heat 
of combustion Q. We may therefore write 

(55) ^ = Eq[i-0)""'], 

and the work due to the combustion of any quantity w of the substance 
is, evidently 

(56) ^ = EQo.[l-0y^']. 

3. Maximum worlc or potential of an explosive suhstance. — If v increases 
indefinitely, the limiting value of the work is, by (55), 

(57) . // = E Q. 

Consequently, the irorlx due to the indefinite expansion of unit of 
weight of a hurned explosive is equal to the product of the mechanical 
equivalent hy the heat of combustion. 

This maximum work we shall call the potential of the exi)]osive sub- 

It is important to notice that the exi)ression /< = E Q of the maximum 
work that unit of weight of an explosive can produce is independent of 
the phenomena of dissociation ; and remains the same in the case Avhere 
the specific heat of the products of combustion vary during the exi^an- 
sion, in consequence of sudden changes in the chemical state of these 
products. In fact, from the equation, 

^cdT -\- pdv = O, 
which represents an infinitely small adiabatic transformation (chai>ter 
I, No. 10), we derive, in all cases where the temperature varies from 
To to absolute zero, 

p dv = ^ I c dT, 

To ■ ' 

and the integral / c dT represents, in all cases, the total heat given 

out by the products in j)assing from zero to To ; that is to say, the heat 
of combustion Q. 

In equations (55) and (56), it is assumed, on the contrary, that the 
ratio n of the two specific heats remains sensibly constant during the 
expansion that is considered. 

4. Potentials of some explosive substances.— The following table ex- 
hibits the values of the potentials of some powders and other explosives. 
These have been calculated by multiplying the experimental heats of 
combustion (chapter II, ^o. 9) by 436. The products being divided by 
1,000, the figures of the table are tonne-meters. 


Name of explosive. 


Sporting powder ' 370 

Cannon powder 347 

Fine-grained powder, called B 337 

Powder of commerce 321 

Ordinary blasting powder ^ 267 

Chloride of nitrogen -, i 144 

Nitro-glycerine - - - 778 

Gun cotton 489 

Picrate of potassium -366 

55 parts of picrate of potassium and 45 of saltpeter 420 

Equal parts of picrate and clilorate of potassium ... I 534 

5. Hypothesis of Bunsen and Schischkoff. — In their chemical theory of 
the combustion of gunpowder, Bunsen and Schischkoff have calculated 
in a different way the maximum work of the powder with which they 
experimented. They made the calculation as though the solid residues 
found after the cooling remained solid throughout the phenomena, and 
kex)t their initial temperature ; thus evaluating only the work of the 
adiabatic expansion of the permanent gases. 

In this hypothesis, formula (53), for the work, is replaced by the fol- 
lowing : 

<f = e E c, (To - T), 

where e is the weight and c the specific heat of the gaseous products of 
the combustion of a kilogram of the substance. 

But we have always c To = Q, Q being the heat of combustion and o 
the specific heat of the entirety of the products. 

We may, then, write ' ■ 

and, in particular, if T tends towards zero, we have the limiting value, 

fe = ^> E Q. 

According to this hypothesis, then, the value of the potential is a 
fraction of what we have found. For example, for sporting powder, the 

ratio - differing little from unity, and e being J, about, we find from the 


corresponding value of E Q, /?- = 120 tonne-meters, about. 

This manner of calculating the work of explosive substances appears, 
however, hardly admissible. Bunsen and Schischkoff themselves ad- 
mit that the volatilization of the solid residues at the temperature of 
5781 3 


combustion, if not certain, is, at least, highly probable ; and, if it be 
true, as these chemists incline to think, that the resulting vapors have 
a tension which may be neglected in comparison with those of the per- 
manent gases, the heat which they give out in cooling to the mean tern- 
perature of the mixture is none the less transformed into work. 

It is not reasonable that these products should keep the same tem- 
perature during the whole expansion ; and if we admit, on thecontrary^ 
that, vaporized, or even solidified, they have at each instant the same 
temperature as the gas, we must adopt the value of the potential, whick 
■we have already laid down. 

In fact, in this second hypothesis the expansion of the gases ceasea 
to be adiabatic, since they absorb the heat given out by the other prod- 
ucts in cooling to the mean temperature of the mixture. If we call e and 
1 — € the relative proportions of the gaseous and non -gaseous products, 
and Ci and c^ the respective specific heats, 

C = e Ci + (1 — e) C2 

represents the mean specific heat of the mixture. 

In cooling from To, the initial temperature, to T, the temperature after 
expansion, the products give up a quantity of heat represented by 5 = 
(1 — e) C2 (To — T) ; and equation (10) applied to a weight e of gas passing 
from To to T, gives 

(l-e)c, (To-T) = e C, (T~To) 4- ~y 

whence we derive 

<? = E [ (1 — e) C, + e Ci] [To — TJ, 

that is to say — 

<f = Ec(To— T), 

an expression identical with (53), and leading, like it, to the value of 
the potential, h = EQ. 

6. First approximation to initial velocities. — Formula (56) gives a first 
approximate determination of the initial velocity of a projectile. 

If we suppose that the combustion of the powder is instantaneous, 
and if we neglect the loss of heat to the walls of the gun, this formula 
will express the work done by the gas in terms of the energy of the pro- 

We have thus — 




V is the initial velocity, 

m the mass of the projectile, 


w the weight of the charge of powder, 

h the potential of the powder, 

To the volume of the powder chamber, 

V the total capacity of the bore, 

n the mean value of the ratio of the two specific heats of the products 

of combustion. 

For this las't element we may adopt, following what has already been 
laid down (chapter II, Ko. 17), the approximate value 

(59) n = 1 +~^ 

in which s is the weight of gas yielded by the combustion of a kilo- 
gram of the powder used. 
From (58) we have 

-c-^y ['-(?)*-']'■ 

as the expression for the initial velocity, if the combustion is instantane- 
ous and no heat is lost. 

We shall show- how this formula may be modified to take account of 
the progressive burning of the grains and of the loss of the heat of the 
products of combustion to the walls of the gun. 


By Professor J. M. Rice, U. S. Navy. 

The values of (d and io' derived in the following note are given in 
Major Sladen's Principles of Gunnery, pages 74 and 75, London, 
1879, with a reference to Mr. W. D. Niven's paper On the Trajec- 
tories of Shot, published in the Proceedings of the Royal Society, No. 
181, Vol. XXVI. 

The expressions for cp and ip' however, as obtained in Mr. Niven's 
paper, differ slightly from those quoted by Major Sladen and derived 

The method employed in deriving these formulas j is similar to 
that of Mr. Niven. 

Putting (p-=La — 4' , equation (4), page 70, becomes 

m — 3 tan a -\- tan'a — 3 tan (« — v'') — tan^ (« — <r'') . 

Expanding the last two terms as functions of c'' bj^Maclaurin's 
Theorem, we have 

—J 5- HZ — 3 sec-a . ^'' + . . . 

up g i_ ' 

-(- 3 tan^a . sec^a . 4' -\- terms involving c''^ L 
1 1 c r 

+ terms involving squares and higher powers of v'' • 

To obtain an approximate value of (1< , we omit all terms contain- 
ing its square and higher powers ; whence 

g cos% /I 1 \ • 


Again, since q is the value of u when c^^Z/^ [see equation (5), 
page 70], we have from (a) 

q COS*« /I 1 \ 

whence 4' = -^ ^ (« — ft) . (^) 

The X Integral. 
In equation (c) , page 73, putting r =:: cv"^ =: cu^ secV, we have 

By Maclaurin's Theorem we have 

cosV "=■ cos^(a — V') =z: cos-a -[- sin 2a . d> -(- terms in v''^ ; (s) 

omitting the terms containing powers of 4> higher than the first and 
substituting in (o) , 

cosVy. f^i' iu , sin 2a /*^ ^ eZw 

Introducing the value of d> from {j~) , we have 
cosV. P^' du , (a — ;5) sin 2a 


Whence, integrating, 

COSV. /1 _ 1\. (g — ;S) sin 2a fJ^/J:^ 1^\ 

V ^/ f ) 

and, i)uttiiia; ^= , 

X= -f- <! cos'a + sin 3« . (« - /5) ^' 'J^^' "!'' P' ^ 

\9 P J 

It is now necessary to obtain an approximate value of the fraction 
_ f "^ cf-g "^ qif f __ (f "^ qp "^ f _ p'' -\-2pq-\r'^t 

\t f) \q-^qp^p') 


p — 1 -4- 1 

Put I =z - — r~^ » whence p = — — '— 7- g . We now substitute this 

value of p in the expression for / and omit r in the result, since I 
is a small quantit}^ 


_ (l + 0- + 2(l-r) + 3(l-0^ 

-4[i + o= + (i-r-) + (i-OT^ 

and, omitting /-, we have approximately 

^~ 4(3 + ^) — 6 * 
Whence Xz= -y | cosV. + sin 2a . (a — ;-) ^^^ | . (ri-) 

Comparing the equations (s) and (r^) , it is e^^dent that the ex- 
pression in braces may be put equal to cosV » if 
3-2Z «-/5 , ^ , ,, 

- 2 ^3(p + ^)^^- ''^- 
That is, we have 

T7ie Y Integral. 
In equation (cZ), page 73, putting^ secV we have 

sin (f cos ^ — ^ . (^) 

Expanding by Maclaurin's Theorem, we have 

sin (f cos (p z=z sin (a — c'') cos (a — (l>) zz: 

sin a cos a — cos 2a . d' -\- terms in c''-. 

Applying to this integral the same process we applied to (0) , we 
have _ 
sin c> cos 

in which <p and Q have the same values as in the X integral. 


The Time Integral. 
Putting r = cu^ secV ^^ equation (i), page 72, we derive 

du , ^ 1 P^ , du 

dtz=. — — = ; whence T-:=. — | cos^c? — - . 

cir ^Qoro cos (p c Jq "^ 

Substituting the value of cosV from equation (e) 

„ cosV. P du I sin 2a /*^ du 

I =Z I ^ -] I (p .■ r • 

C Jq W G Jq U^ 

Integrating and introducing the value of 4' from the equation (j) 
cosV. 1 /j^_ 1\_| (^ — /^) sin 2« PV du^ 1__^\ 

_ cos^g 1/1 1\ (« — jg) sin2a rj^/J^__ j^\ 

— c 2 v^^ ^^'/ cT- L^L^^^' ^' ^ 

\q' f ) 
whence, putting Q^:=z~--l -^ — ~^ )' ^® bave 



3 v?' p°-J\g' p' J 

Denoting by /' the fraction in the right hand term, and reducing 

\f p J p \ q p" J 




Putting . = f^. whe.ce^ = ^., we have 

iS — S^+Syi^l __ InL (omitting V and V) . 
whence / = iqT^jZ^ 6 

Hence T=^{ cos=« + ^i" ^a . (« - « • -q- ] ' 
on comparing this equation with (0 , it is obvious that we can 
express T in the form ^ ^ ^ 

y^^eosV, if 7 = «-^(«-« = ''- V + T^"-^^' 


?>' = -^^ 6 p + 9 "- 

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